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Stability of Runge-Kutta methods for the generalized pantograph equation. (English) Zbl 0943.65091

Stability properties of Runge-Kutta (RK) methods applied to nonautonomous neutral dealy-differential equations with a constant delay of the form \[ u'(t)- Ku'(t-\tau)= e^t Lu(t)+ e^tMu(t-\tau),\quad t>0,\tag{\(*\)} \] are studied. Equation \((*)\) is obtained from the autonomous generalized pantograph equation \[ v'(t)- Kv'(\alpha t)= Lv(t)+ Mv(\alpha t),\quad 0<\alpha< 1,\tag{\(**\)} \] by the change of the independent variable \(u(t)=v(e^t)\). When the RK matrix is regular, it is shown that the stability properties of the RK method for \((*)\) may be derived from those for a difference equation. Further, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to \((**)\).

MSC:

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34K40 Neutral functional-differential equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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